Integer shifts on the characteristic exponents of Fuchsian systems preserving their monodromy are
called the Schlesinger transformations [1]}. In [2]}, the authors
studied the Schlesinger transformations of monodromy preserving deformation including irregular
singular case and showed that the ratio \(\tau ^{\prime }/\tau \)  is a determinant of a matrix whose elements
are given in terms of the coefficients of fundamental solutions,
where \(\tau \)  is the Jimbo–Miwa–Ueno tau function
introduced in [3]} and \(\tau ^{\prime }\)  is a Schlesinger transform of the original tau function. We also mention a recent work [4]} where the authors presented
another determinant formula for the ratio of \(\tau \) -functions by using relations between Hermite's two approximation problems and particular Schlesinger transformation.
